Theorem f1oen2g

Description: The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 10-Sep-2015)

	Ref	Expression
Assertion	f1oen2g	|- ( ( A e. V /\ B e. W /\ F : A -1-1-onto-> B ) -> A ~~ B )

Proof

Step	Hyp	Ref	Expression
1		f1of	|- ( F : A -1-1-onto-> B -> F : A --> B )
2		fex2	|- ( ( F : A --> B /\ A e. V /\ B e. W ) -> F e. _V )
3	1 2	syl3an1	|- ( ( F : A -1-1-onto-> B /\ A e. V /\ B e. W ) -> F e. _V )
4	3	3coml	|- ( ( A e. V /\ B e. W /\ F : A -1-1-onto-> B ) -> F e. _V )
5		simp3	|- ( ( A e. V /\ B e. W /\ F : A -1-1-onto-> B ) -> F : A -1-1-onto-> B )
6		f1oen3g	|- ( ( F e. _V /\ F : A -1-1-onto-> B ) -> A ~~ B )
7	4 5 6	syl2anc	|- ( ( A e. V /\ B e. W /\ F : A -1-1-onto-> B ) -> A ~~ B )